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Let $(V,\mathbb{F})$ be a vector space (with $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$). We say that the inner product of vectors $\vec{v_1}$ and $\vec{v_2}$, written $\langle v_1,v_2 \rangle$, is a function that takes any two vectors to a number in $\mathbb{F}$ and has the following additional properties:
1. (Symmetry or Conjugate Symmetry) When dealing with only real numbers, the following symmetry formula must hold:
$$\langle x,y \rangle = \langle y,x \rangle,$$
and when dealing with complex numbers, the following conjugate symmetry formula holds:
$$\langle x,y \rangle = \overline{\langle y,x \rangle},$$
where $\overline{\langle y,x \rangle}$ denotes complex conjugation.
2. (Linearity in the first argument) The following formula holds for all scalars $\alpha,\beta$ and vectors $\vec{x},\vec{y},\vec{z}$:
$$\langle \alpha \vec{x}+\beta \vec{y},\vec{z} \rangle = \alpha \langle \vec{x},\vec{z} \rangle + \beta \langle \vec{y},\vec{z} \rangle.$$
Note in the case of real inner products we can factor out of the second term as
$$\langle \vec{x},\alpha\vec{y} \rangle =\alpha \langle \vec{x},\vec{y} \rangle$$
but if we are dealing with a complex inner product, factoring out of the second term results in a conjugate factor:
$$\langle \vec{x},\alpha \vec{y} \rangle = \overline{\alpha} \langle \vec{x},\vec{y} \rangle.$$
3. (Positive definiteness) It is always true that
$$\langle \vec{x},\vec{x} \rangle \geq 0$$
and $\langle \vec{x},\vec{x} \rangle=0$ if and only if $\vec{x}=\vec{0}$.
If $(V,\mathbb{F})$ is a vector space and $\langle \cdot,\cdot \rangle$ is an inner product on $(V,\mathbb{F})$, then we say that $(V,\mathbb{F},\langle \cdot,\cdot \rangle)$ is an inner product space.
Examples of inner product spaces
1. Let $\vec{x},\vec{y} \in \mathbb{R}^{n \times 1}$ with $\vec{x}=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}$ and $\vec{y}=\begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix}$. Then the following formula defines an inner product:
$$\langle \vec{x}, \vec{y} \rangle = x_1y_1 + x_2y_2 + \ldots + x_ny_n = \displaystyle\sum_{k=1}^n x_ky_k.$$
2. Let $\vec{p},\vec{q} \in \mathbb{P}$ (the vector space of polynomials). The following formula defines an inner product:
$$\langle \vec{p},\vec{q} \rangle = \displaystyle\int_0^{\infty} \vec{p}(x)\vec{q}(x)e^{-x} dx.$$
3. Let $f,g \in C[0,1]$, the functions continuous on the interval $[0,1]$. The following formula defines an inner product:
$$\langle f,g \rangle=\displaystyle\int_0^1 f(x)g(x)dx.$$
4. Let $\{a_k\}_{k=0}^{\infty}, \{b_k\}_{k=0}^{\infty} \in \ell^1(\mathbb{R})$, the set of sequences $\{x_k\}$ such that $\displaystyle\sum_{k=0}^{\infty} |x_k| < \infty$. The following formula defines an inner product:
$$\langle \{a_k\},\{b_k\} \rangle = \displaystyle\sum_{k=0}^{\infty} a_kb_k.$$
5. Let $z_1, z_2$ be complex numbers with $z_1=a+bi$ and $z_2=c+di$ and $i^2=-1$. The following formula defines an inner product:
$$\langle z_1,z_2 \rangle = z_1 \overline{z_2},$$
where $\overline{z_2}=c-di$ (called the complex conjugate of $z_2$).
